If $f: R \rightarrow R$ is defined as $f(x)=\frac{2020^x}{2020^x+\sqrt{2020}}$,$\forall x \in R$,then $\sum_{r=1}^{4039} 2 f\left(\frac{r}{4040}\right)=$

Let $A = \{1, 3, 4, 6, 9\}$ and $B = \{2, 4, 5, 8, 10\}$. Let $R$ be a relation defined on $A \times B$ such that $R = \{((a_1, b_1), (a_2, b_2)) : a_1 \leq b_2 \text{ and } b_1 \leq a_2\}$. Then the number of elements in the set $R$ is

The number of roots of the equation $\sqrt{2}+e^{\cosh^{-1} x}-e^{\sinh^{-1} x}=0$ is

Answer the following by appropriately matching the lists based on the information given in the paragraph.
Let $f(x) = \sin(\pi \cos x)$ and $g(x) = \cos(2\pi \sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in increasing order:
$X = \{x : f(x) = 0\}, Y = \{x : f'(x) = 0\}$
$Z = \{x : g(x) = 0\}, W = \{x : g'(x) = 0\}$
$List-I$ contains the sets $X, Y, Z$ and $W$. $List-II$ contains some information regarding these sets.

$List-I$	$List-II$
$(I) X$	$(P) \supseteq \{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\}$
$(II) Y$	$(Q) \text{ an arithmetic progression}$
$(III) Z$	$(R) \text{ NOT an arithmetic progression}$
$(IV) W$	$(S) \supseteq \{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\}$
	$(T) \supseteq \{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\}$
	$(U) \supseteq \{\frac{\pi}{6}, \frac{3\pi}{4}\}$

$(1)$ Which of the following is the only $CORRECT$ combination?
$(1) (II), (R), (S)$ $(2) (I), (P), (R)$ $(3) (II), (Q), (T)$ $(4) (I), (Q), (U)$
$(2)$ Which of the following is the only $CORRECT$ combination?
$(1) (IV), (Q), (T)$ $(2) (IV), (P), (R), (S)$ $(3) (III), (R), (U)$ $(4) (III), (P), (Q), (U)$

Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{1, 2, 3, 4, 5, 6\}$. Then the number of functions $f: A \rightarrow B$ satisfying $f(1) + f(2) = f(4) - 1$ is equal to

If the graph of a non-constant function $f(x)$ is symmetric about the point $(3,4)$,then the value of $\sum\limits_{r = 0}^6 {f(r) + f(3)}$ is equal to